MMN-4255
Note on the moment generating function of the multivariate normal distribution
Kenichi Hirose;Abstract
We present a formula for the derivatives of the moment generating function of the multivariate normal distribution. We formulate it in terms of the summation of the contractions by pairings, which encodes a combinatorial computation procedure. We give two applications. First, we provide a simple proof of Isserlis’ theorem and derive a
formula for the moments of the multivariate normal distribution. Second, we prove a formula for the moments of the product of a finite number of correlated normally and lognormally distributed random variables.
Vol. 25 (2024), No. 1, pp. 287-300